Yes you can give your goat a bath like you would do for horses. They dont like it too well but during the hot weather they would appreciate it. I had bathe and clipped Nubian goats for 4H when I was a kid.
House training goats would be like trying to keep them in a single wire electric wire to keep them in....they will definately get on everything and anything. Not a good idea. Keeping them from soiling is not an easy feat either forgo putting a goat in house or get a diaper for it.
By the way, NO billy goats! They can stink high heaven and very dominant in some cases. That is why we never had billy goats on our farm, we just take the girls for stud services down the road aways, and bring them back after a period of time. Works well for us!